$E_0$-semigroups: around and beyond Arveson's work

TL;DR

This paper reviews the theory of $E_0$-semigroups, highlighting Arveson's foundational work and recent advances in type II and III classifications, along with a note on noncommutative Poisson boundaries.

Contribution

It provides a comprehensive account of the development of $E_0$-semigroups, including recent classifications and insights into noncommutative Poisson boundaries.

Findings

Overview of Arveson's contributions to $E_0$-semigroups

Recent classification of type II and III $E_0$-semigroups

Insights into noncommutative Poisson boundaries

Abstract

We give an account of the theory of $E_0$-semigroups. We first focus on Arveson's contributions to the field and related results. Then we present the recent development of type II and type III $E_0$-semigroups. We also include a short note in Appendix, based on Arveson's observation, on noncommutative Poisson boundaries.